How to derive it
- Let θ = 18°, so 5θ = 90° and 2θ = 90° − 3θ.
- Then sin 2θ = cos 3θ: 2 sin θ cos θ = 4cos³θ − 3 cos θ.
- Divide by cos θ: 2 sin θ = 1 − 4 sin²θ, so 4 sin²θ + 2 sin θ − 1 = 0.
- Solve the quadratic and take the positive root: sin 18° = (√5 − 1)/4.
- cos 36° = 1 − 2 sin²18° = (√5 + 1)/4.
- sin²36° = 1 − cos²36° = 1 − (6 + 2√5)/16 = (10 − 2√5)/16.
- So sin 36° = √(10 − 2√5)/4.
All trig ratios of 36°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 36° | √(10 − 2√5)/4 | 0.5878 |
| cos 36° | (√5 + 1)/4 | 0.8090 |
| tan 36° | √(5 − 2√5) | 0.7265 |
| cosec 36° | 1 ÷ sin 36° | 1.7013 |
| sec 36° | 1 ÷ cos 36° | 1.2361 |
| cot 36° | 1 ÷ tan 36° | 1.3764 |
Exact values of special angles
For whole-degree values from 0° to 90°, see the trigonometry table.
Questions students ask
What is the value of sin 36°?
√(10 − 2√5)/4, which is about 0.5878.
What is sin 36° in radians?
36° is π/5 radians, so sin(π/5) = √(10 − 2√5)/4.
Is sin 36° equal to cos 54°?
Yes. sin θ = cos(90° − θ), so sin 36° = cos 54°.
How do you find sin 36° without a calculator?
Let θ = 18°, so 5θ = 90° and 2θ = 90° − 3θ. Then sin 2θ = cos 3θ: 2 sin θ cos θ = 4cos³θ − 3 cos θ. Divide by cos θ: 2 sin θ = 1 − 4 sin²θ, so 4 sin²θ + 2 sin θ − 1 = 0. Solve the quadratic and take the positive root: sin 18° = (√5 − 1)/4. cos 36° = 1 − 2 sin²18° = (√5 + 1)/4. sin²36° = 1 − cos²36° = 1 − (6 + 2√5)/16 = (10 − 2√5)/16. So sin 36° = √(10 − 2√5)/4.